Goku didn't have SSJ3?

[Papasmurf]

Let's say Galu wasn't bullshitting about SSJ2 being his highest power. How does this change the events of the Boo arc and DBS?

 

[Captain Cadaver]

With it being a certainty he'd be erased from existence if not being able to reason with Boo, Goku would just use Shunkan Ido to get the Dragon Radar himself and focus on helping the boys fuse. With Babidi still around, Gotenks probably gets killed on Babidi's order when he goes to confront Boo, Mr. Satan never gets the chance to befriend Boo and Ultimate Gohan ends up being the one to save the day. If Earth is completely destroyed before he can get there or Dende is killed, then Goku tells Kibito to take Gohan to New Namek to use Porunga to restore everything.
When Beerus wakes up, assuming he eventually gets sick of waiting around for SSG despite Boo's absence, then they use the Dragon Balls to bring Tarble to Earth for the ritual and Vegeta takes Goku's place. When Freeza comes to Earth, it's difficult to say if Vegeta would've unlocked SSB without a sparring partner like Goku and it's doubtful Vegeta's actions against Freeza would change much, so Freeza wins either way. Whether or not he'll continue his rule uninterrupted though depends on if he still has his forces visit Vampa and if he ever has the incentive to kill Paragus.

 

[Hector]

Goku probably goes to fight Buu anyway and stalls him a little with SSJ2, teleporting when things get out of hand. The rest of the events happen as we know until the fight in the ROSAT. This time, Super Buu is the only one to escape. Ultimate Gohan eventually gets absorbed. Vegetto is absorbed later and Gohan is released. With no SSJ3, Vegeta has to accept to fuse with Goku one more time to fight Kid Buu (through fusion dance). Everything gets fixed with the Dragon Balls. DBS goes as we know it.